The real Hardy space $H^p$ defined by a radial maximal function

Fix $n\ge 1$, $0<p<\mathrm{\infty }$ and an admissible kernel $\phi \in \mathcal{S}({\mathbb{R}}^n)$ with ${\int }_{{\mathbb{R}}^n}\phi \mathrm{\ne }0$, and let $M_{\phi }^{0}f$ be the radial maximal function of Radial and nontangential maximal functions of a tempered distribution. The real Hardy space is $H^p({\mathbb{R}}^n)=\{f\in {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n):M_{\phi }^{0}f\in L^p({\mathbb{R}}^n)\},$ with the functional $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}:=\mathrm{\parallel }{M}_{\phi }^{0}f{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\mathrm{.}$ Here $L^p$ is the quotient by almost-everywhere null functions of The space $L^p(\mu )$ as the quotient by null functions with the complex scalar conventions of Complex Lp classes and Euclidean test-function conventions, and $M_{\phi }^{0}f$ is the Borel measurable extended-real function supplied by Measurability and lower semicontinuity of the smooth maximal functions; the membership condition includes that $M_{\phi }^{0}f$ is finite almost everywhere and that its class lies in $L^p$. Since $M_{\phi }^{0}f\ge 0$, the functional takes values in $[0,\mathrm{\infty }]$ and is finite on $H^p$; the zero distribution lies in $H^p$ with $\mathrm{\parallel }0{\mathrm{\parallel }}_{H^p}=0$. Under Countable Choice, for $p\ge 1$ the functional $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H^p}$ is a norm. For $0<p<1$ it is $p$-subadditive. No Banach-space duality of $H^p$ with a normed dual is asserted here below $p=1$. The kernel $\phi$ is held fixed in the definition; the maximal-characterisation theorem on this page shows that different admissible kernels give the same space with equivalent quasi-norms, so that the notation $H^p({\mathbb{R}}^n)$ does not depend on the choice up to equivalence. No choice principle is used in the definition itself.

The set defining $H^p$ is specified without a choice principle. Under Countable Choice, positive definiteness holds for every $p>0$. If $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}=0$, then $M_{\phi }^{0}f=0$ almost everywhere. Lower semicontinuity of the radial maximal function makes it identically zero: if it has a positive value, one of its open strict superlevel sets contains a Euclidean ball, which has positive measure under Countable Choice by Euclidean balls have positive finite Lebesgue measure. Thus $f\ast {\phi }_t=0$ for every $t>0$ at every point. Apply Schwartz approximate identities converge in the sense of tempered distributions to $\mathrm{\Phi }=\phi \mathrm{/}(\int \phi )$ to conclude $f=0$ in ${\mathcal{S}}^{\mathrm{\prime }}$. The lower-semicontinuity input is Measurability and lower semicontinuity of the smooth maximal functions, and the approximate-identity limit is as stated. For $p\ge 1$, homogeneity and the triangle inequality follow from $M_{\phi }^0(f+g)\le M_{\phi }^{0}f+M_{\phi }^{0}g$ and the complex $L^p$ norm properties Complex Holder, Minkowski, and the quotient norm, so the functional is a norm.

For $0<p<1$, pointwise sublinearity and $(u+v{)}^p\le u^p+v^p$ for $u,v\ge 0$ give $\mathrm{\parallel }f+g{\mathrm{\parallel }}_{H^p}^p\le \mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}^p+\mathrm{\parallel }g{\mathrm{\parallel }}_{H^p}^p$. The $H^p$ membership definition itself uses no choice principle.